That exists, it's called a DMCA 512 counternotice; if you're particularly monied you can even sue under 512(f) for perjury. The problem is that the videos are actually infringing, but the owner of the content does not want to sue.

Under very basic principles of law, only the owner or exclusive licensee of a copyright has standing to sue for copyright infringement. Furthermore, copyright law does not obligate copyright owners sue or license like trademark does. Therefore, for uses which are inconvenient[0] to sell a license for, but not damaging enough to go to court, copyright owners will often tacitly permit the use by simply failing to enforce their rights.

The problem is that courts have a very high bar to recognize tacit permission as a license. It's not impossible; there are some famous examples of 'implied license', but no competent lawyer would actually recommend you go to court and claim such a thing. One particular complication would be that if, say, you sued Fake Nintendo, and claimed fair use as a rationale for using Real Nintendo's content, Real Nintendo might want to actually sue you just to kill the fair use claim[1].

Just as an example of how complicated tacit permission can get:

Bungie's Destiny 2 is a perpetually updated "live service" game with an ongoing policy of removing content to keep download sizes reasonable[2]. As a result, there is music in the game that is no longer accessible. Bungie does not want people uploading the game soundtrack to YouTube, but they also don't want to turn that removed music into lost media, so they had a policy of not taking down "music archivists" that only uploaded the removed content.

One of the YouTubers that got taken down for reuploading live Destiny 2 music got pissed about it and started filing fraudulent DMCA takedowns in Bungie's name to music archivists. Bungie tried to get in contact with YouTube to have the fraudulent takedowns removed, but it took over a week of PR damage to everyone involved (and, if I remember, actually suing the idiot kid that did this) before YouTube would restore the videos.

If there is one thing that is badly drafted (and not just irritating) about the current DMCA 512 system, it's that there is no procedure for third-party counter-claimants to challenge fraudulent or mistaken claims. However, the current mechanisms of copyright make that impossible to provide. There is no database of who-owns-what and who-licensed-what; rights owners do not want such a database to exist; and it is entirely possible for multiple parties to have standing to sue the same person for the same act of infringement on the same work. Under regular copyright law, if Nintendo wants to sue you for, say, using the officially-licensed Mario DLC in your Minecraft streams, Microsoft can't intervene and stop them on the basis that they own Minecraft. How, exactly, should YouTube proceed if they have two parties swearing under oath conflicting information, and not obeying the right one puts you on the hook for billions of dollars in copyright liabilities? The current system is designed to make it easy to cheaply operate social media, not to actually be fair to its users or to stop online censorship.

[0] Reasons for this inconvenience can include:

- The transaction cost of negotiating a watertight contract for a very small deal. Generally speaking you don't want to make deals with understandable / 'plain language' licensing terms for the same reason why web browsers don't have an API to load unsigned arbitrary kernel modules from third-party servers.

- The licensing in question being contrary to exclusive licensing arrangements with other companies - though exclusive licensing contracts can also mandate the licensor or licensee enforce each other's rights to prevent this sort of thing

[1] In general, common law mechanisms like fair use create an incentive to sue, which is a very bad thing for people who don't like getting sued.

[2] This is a terrible policy, but the policies of console manufacturers require you to ship games as packages, so you couldn't just stream in assets as needed.